LS30A_W24_Exam_KEY_PDF

E. Deeds LS 30A, Lec. 1 Friday Feb 16, 2024 Midterm Name: TA Name: Section Number : Student ID: Instructions: Do not open this exam until instructed to do so. You will have 110 minutes to complete the exam. Please print your name and your student ID number above. Also, Please print your name and UID in the top right-hand corner of every page. You may not use books, notes, or any other material to help you. You may use a scientific calculator on this exam. Please make sure your phone is silenced and stowed with your other belongings at the front of the room. You may use any available space on the exam for scratch work, including the backs of the pages. If you need more scratch paper, please ask one of the proctors. By signing below, you a ffi rm that you have neither given nor received unautho- rized help on this exam and that you have read and agreed to the instructions detailed above Signature: Exam key

UID: 1. (9 points) You are modeling a new video game. People can play this game for free, but if they pay for the game they get extra rewards. Everyone who starts playing the game starts as free-to-play, but some of them eventually decide to pay. To model this system, call the number of free-to-play players F and the number of paid players P . The game developers continuously add content to the game; let’s quantify the amount of content in the game with the variable C , as measured in gigabytes of in-game assets. Each rate is per month. Consider the following assumptions: • A constant number t players per month join the game. Remember, players always start as free-to-play. • The amount of content in the game influences players’ decision to spend money. As such, the per capita rate at which players switch from free-to-play to paid is proportional to the amount of content, with proportionality constant s . • Some free-to-play players get bored and decide to quit the game. Model this as a constant per capita rate q of players quitting the game. • Paid players sometimes decide to go back to being free-to-play, but the more content there is in the game, the less likely they are to do so. The per capita rate at which paid players go back to being free to play is thus inversely proportional to the amount of content in the game, with proportionality constant r . • The more paid players there are in the game, the more money the developers have to create new content. The amount of content in the game thus increases at a rate proportional to the number of paid players, with proportionality constant m . Question 1 continues on the next page. . . key t SFC GF Pt MP

Question 1 continued. . . UID: (b) (2 points) Is there a feedback loop present between the amount of content in the game and the number of paid players? If so, indicate whether that feedback loop is positive or negative . It may be helpful to start by drawing a feedback diagram for this system. (c) (2 points) Say the developers implement a new system in which there are more rewards for paid players. This will increase the rate at which free-to- play players become paid players. How would you model this in the change equations you wrote above? Question 1 continues on the next page. . . key Yes there is a feedback loop between C and P It is a positive feedback loop because increasing P increases C and increasing C increases P There are only positive c f links thus the feedback P loop is positive increase value of parameter S in the term SFC that models free to play conversion M

Question 1 continued. . . UID: This Page Intentionally Left Blank Work on this page will not be graded! Key

Question 2 continued. . . UID: (b) (2 points) What are the units on the parameter u ? (c) (2 points) What is the state space for this system? Define it using the R notation we learned in class. Also, draw a sketch of the state space for this problem (remember to label your axes! ). 3. (8 points) You work at a hospital as a resident in Internal Medicine. A number of your patients on the floor are all su ff ering from the same disease, but thankfully there is a drug that can cure this disease over time. The drug is administered by IV, and, like many drugs, is dosed according to the patient’s weight. In other words, patients don’t all receive the same amount of the drug; patients who weigh more get a higher dose, while patients who weigh less get a lower dose. Answer the following questions based on this situation. Question 3 continues on the next page. . . Key gownton indggat a s semafight Sien anemone Individual monthtindividi monthlownf IR al

Question 3 continued. . . UID: (a) (2 points) There is a table at the nurse’s station where each patient receiv- ing this drug has their weight in kilograms (rounded to the nearest kilogram) recorded. Use the symbol w for this table. Can you describe this table as a function? If so, write out the standard f : {Domain} ≠æ {Co-domain} definition for this function. If not, describe why it is not a function. (b) (2 points) The pharmacy department has a simple formula that allows them to calculate the dose of the drug in milligrams. The formula is d ( m ) = 0 . 02 m , where m is the weight of the patient in kilograms, and d is the resulting dose of the drug. We are using the symbol d for this relationship. Can you describe this table as a function? If so, write out the standard f : {Domain} ≠æ {Co-domain} definition for this function. If not, describe why it is not a function. Question 3 continues on the next page. . . Key yes w patient weight d weigh't dose

UID: 4. (6 points) You are studying a population of Asian small-clawed otters living in southeast Asia. These otters eat small fish and shellfish, and you decide to use a logistic model for the otter population A . You develop the following model: A Õ = 0 . 2 A (1 ≠ A 2000 ) The timescale of this model is months. Answer the following questions: (a) (5 points) When you first arrive at the field site where you are studying the otters, you find that there are 1000 otters present. Call the time you arrived t = 0 . Use Euler’s method with a stepsize of Δ t = 0 . 1 to predict how many otters you will have in 0 . 2 months. (b) (1 point) A major development project is being planned for the area, and environmental studies have shown that, if it proceeds, the maximum sus- tainable size of the otter population will be cut in half. How would you change your model to reflect that scenario? Key to t 1000 Afgan pg At t At tote 100 0.1 10 11000 100 10 1010 0.1 Af 010 At At Ot Attot 0.211010311 1 9 99.99 10.1 9.999 1010 202 l 0.505 20210.495 9.999 1019.999 99.99 0.2 At fractional and 1019.9995 m't reduce the carrying capacity 2000 by half so 2000 becomes 1000

UID: 5. (9 points) You are studying a black bear population ( B ) in the Pacific North- west. These bears feed on salmon ( S ). Based on your measurements of the two population sizes B and S , you obtain the following trajectory for this system: Note that the times (in years) at which each measurement was made are indicated on the graph. (a) (2 points) Explain how the interactions between the bears and salmon cause both these populations to behave in this manner. Question 5 continues on the next page. . . Key for this question Multiple answers were accepted Initially the bean population is low and the salmon population is high As the bears eat the salmon their populationincreases Initially this has little impact on on the salmon population but as the bear population increases eventually predation of salmon by bears leads to acollapse in the salmon population This collapse leads to a subsequent collapse in the bear population due to a lack of salmon for the bears to eat

UID: 6. (4 points) The graph below represents the response of cancer cells to treatment with a drug in a dish. The y -axis is C , the number of cancer cells alive after treatment with the drug, and the x -axis is D , the dose of the drug. Using this graph, answer the questions below: 0 1 2 3 4 5 0 10 20 30 40 50 60 D (Drug dose in μ g) C (# of Cancer Cells) X W Y Z (a) (1 point) Which dashed line above represents a secant line? (b) (1 point) Which dashed line is a tangent line? (c) (1 point) Which dashed line has a slope that would represent the average decrease in the cancer cell population between two points? (d) (1 point) Which dashed line has a slope that would represent the instan- taneous rate of decrease in the cancer population at a point? key Y Z Y z